Abstract.
It seems typical that critically non-recurrent maps, so called Misiurewicz maps, are rare in the parameter space. This was shown by D. Sands in the real quadratic family
f_a(x)=1-ax^2, where 0 < a < 2; In this family a Misiurewicz map f_a has no attracting cycles and the critical point x=0 is non-recurrent. D. Sands showed that the set of parameters a for which f_a is a Misiurewicz map has Lebesgue measure zero.
I will talk about a corresponding result for rational maps on the Riemann sphere. For rational maps, a non-hyperbolic map f is called a Misiurewicz map if f has no parabolic cycles and such that every critical point c on the Julia set is not contained in the omega-limit set of any critical point. In other words, the critical set on the Julia set is non-recurrent. These maps has Lebesgue measure zero in the parameter space of rational maps of any fixed degree d>=2. This will be the main topic of the talk.
If time permits, I will also mention some other nearby results such as the Hausdorff dimension for Misiurewicz maps, measure zero for (second degree) semi-hyperbolic maps studied by Carelson, Jones, Yoccoz (both results a joint project with J. Graczyk) and approximation by hyperbolic maps.